Results 1 to 5 of 5

Math Help - Not completely understanding what analytic means. HELP!

  1. #1
    Newbie
    Joined
    Apr 2014
    From
    Alabama
    Posts
    11

    Exclamation Not completely understanding what analytic means. HELP!

    I understand the following:
    Theorem: Suppose the real functions u(x,y) and v(x,y) are continuous and have continuous 1st order partials in a domain D. If u and v satisfy the CR equations at all points in D, then f(z) = u(x,y) +iv(x,y) is analytic in D.

    Therefore, the CR equations u_{x} = v_{y} and u_{y}=-v_{x} must hold.

    But I don't understand what I'm supposed to conclude when I do the following problem:
    f(z)=|z-10|^2 = y^2 + (x^2 - 10)^2

    When I do the CR equations I get v_{y} = 0 and v_x=0. Does that mean it is only differentiable at (0,0)? What does that say about being analytic? I've seen a few problems like this were some of the CR equations are equal to zero such as f(z) = |z|^2. For that one I understand that limits at different paths are not equal therefore only differentiable at z = 0. Do I need to do different paths to show where it's only differentiable for the problem above or how should I try to tackle this problem? Thank you so much!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,696
    Thanks
    1467

    Re: Not completely understanding what analytic means. HELP!

    "Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at z= z_0, if and only if it is equal to its Taylor's polynomial in some open neighborhood of z_0, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2014
    From
    Alabama
    Posts
    11

    Re: Not completely understanding what analytic means. HELP!

    Quote Originally Posted by HallsofIvy View Post
    "Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at z= z_0, if and only if it is equal to its Taylor's polynomial in some open neighborhood of z_0, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
    Okay, I've read my notes a few more times and I think I'm beginning to understand. For the problem I stated I got u_x = 2x - 20, u_y = 2y, v_x = 0, and finally v_y = 0. Therefore, 2x -20 = 0 and 2y = 0. Therefore, it is only differentiable at (0,0). So how do I jump to the conclusion that it is (or it isn't) analytic?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Apr 2014
    From
    Alabama
    Posts
    11

    Re: Not completely understanding what analytic means. HELP!

    Quote Originally Posted by HallsofIvy View Post
    "Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at z= z_0, if and only if it is equal to its Taylor's polynomial in some open neighborhood of z_0, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
    And by paths I meant taking the limit along the real axis or the imaginary axis or any other point. If both limits aren't the same then the limit of f(z) does not exist.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2014
    From
    Alabama
    Posts
    11

    Re: Not completely understanding what analytic means. HELP!

    I've read my notes over and over again and I'm still stuck. Can anyone help me out?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: March 5th 2012, 04:50 PM
  2. Replies: 0
    Last Post: December 14th 2011, 05:02 AM
  3. Help understanding what this theorem means
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 9th 2011, 12:48 AM
  4. Replies: 3
    Last Post: September 14th 2009, 10:16 AM
  5. Factoring Completely
    Posted in the Algebra Forum
    Replies: 2
    Last Post: May 5th 2008, 04:41 PM

Search Tags


/mathhelpforum @mathhelpforum